CN107703740A - A kind of robust interval sensor fault diagnosis method of bullet train critical system - Google Patents

A kind of robust interval sensor fault diagnosis method of bullet train critical system Download PDF

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CN107703740A
CN107703740A CN201710554307.XA CN201710554307A CN107703740A CN 107703740 A CN107703740 A CN 107703740A CN 201710554307 A CN201710554307 A CN 201710554307A CN 107703740 A CN107703740 A CN 107703740A
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msub
msup
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周东华
张峻峰
何潇
卢晓
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Shandong University of Science and Technology
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Shandong University of Science and Technology
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    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
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Abstract

The invention discloses a kind of robust interval sensor fault diagnosis method of bullet train critical system, belong to field of signal processing, this method includes：Establish bullet train critical system discrete-time state-space model step；Design bullet train critical system robust Residual Generation device step；Design bullet train critical system robust interval Transducer-fault Detecting Method step；Design the separation of bullet train critical system robust interval sensor fault and method of estimation step.The effective guarantee of the present invention practical application request of bullet train critical system interval sensor fault diagnosis.

Description

A kind of robust interval sensor fault diagnosis method of bullet train critical system
Technical field
The invention belongs to field of signal processing, and in particular to a kind of robust interval sensor event of bullet train critical system Hinder diagnostic method.
Background technology
Bullet train be in high ferro technology with the most close system of passenger's relation, its safety, even running be high ferro health, Fast-developing basis and premise.Fault diagnosis technology is the key technology for ensureing bullet train safe operation.At present, arrange at a high speed Car has taken many technical measures and system to improve the reliability of train operation and security.
However, existing bullet train fault diagnosis technology has many limitations, such as mostly simple threshold value compares, and arranges in pairs or groups Tactic operating procedure, without the redundancy fully excavated available for fault diagnosis；Diagnostic function heavy dependence failure code The richness of table, can only existing most common failure in DTC table, can not diagnose caused new in train actual motion Failure；Reply situation considers permanent fault more, and does not consider the interval sensor fault as caused by the factors such as electromagnetic interference.
Based on the above situation, a kind of robust interval sensor fault diagnosis method of bullet train critical system is needed badly, with Realize that fault diagnosis system carries out real-time online diagnosis to bullet train critical system interval sensor fault.
The content of the invention
For above-mentioned technical problem present in prior art, the present invention proposes a kind of Shandong of bullet train critical system Rod interval sensor fault diagnosis method, it is reasonable in design, the deficiencies in the prior art are overcome, there is good effect.
To achieve these goals, the present invention adopts the following technical scheme that：
A kind of robust interval sensor fault diagnosis method of bullet train critical system, specifically comprises the following steps：
Step 1：Establish bullet train critical system discrete-time state-space model and identification model parameter
Wherein,Respectively system mode, control input, measurement output； Respectively process noise, measurement noise；For sensor fault； For systematic parameter； For parameter uncertainty；Meet following condition：Original state x (0) average, covariance, second moment are respectivelyP0, Σ0；Make an uproar Sound w (k), v (k) average are zero, and covariance matrix is respectively Σw(k), Σv(k)；Parameter uncertainty Aδ(k),Bδ(k),Cδ(k) Average be zero, covariance matrix is respectively
Step 2：Bullet train critical system robust Residual Generation device is designed, is specifically comprised the following steps：
Step 2.1：Off-line calculation gain matrix K (k)
K (k)=G (k) Cc(k)TQ(k)-1, (2)；
Wherein,
Rapid 2.2：In line computation robust residual error r (k)
Wherein,
Step 3：Bullet train critical system robust interval Transducer-fault Detecting Method is designed, specifically includes following step Suddenly：
Step 3.1：Calculate intermittent fault detection statistic TD(k)
Step 3.2：Set intermittent fault detection rate of false alarm Pfa
Pfa=αfa, (12)；
Step 3.3：Design intermittent fault detection threshold value JD
Wherein,
Step 3.4：Intermittent fault detection is carried out according to following criterions
If TD(k-1)≤JD,TD(k) ＞ JD, then the k moment break down, fault warning indicatrix Ia=1；
If TD(k-1) ＞ JD,TD(k)≤JD, then k moment failure vanishes, trouble shooting indicatrix Ir=1；
Correspondingly, the fault warning time of i-th intermittent fault and trouble shooting time are respectively
kalarm,i=min (k | TD(k) ＞ JD,k≥krelease,i-1+1), (15)；
krelease,i=min (k | TD(k)≤JD,k≥kalarm,i+1), (16)；
Step 4：Design the separation of bullet train critical system robust interval sensor fault and method of estimation, specifically include as Lower step：
Step 4.1：Calculate intermittent fault separation statistic TI(i,j)
Wherein,
L(i,j)11=G (i, j)11 TT(i)-1G(i,j)11, (18)；
L(i,j)12=G (i, j)11 TT(i)-1G(i,j)12, (19)；
L(i,j)21=G (i, j)12 TT(i)-1G(i,j)11, (20)；
L(i,j)22=G (i, j)12 TT(i)-1G(i,j)12, (21)；
d(i,j)1=G (i, j)11 TT(i)-1r(kalarm,i,krelease,i-1), (22)；
d(i,j)2=G (i, j)12 TT(i)-1r(kalarm,i,krelease,i-1), (23)；
r(kalarm,i,krelease,i- 1)=[r (kalarm,i)T r(kalarm,i+1)T … r(krelease,i-1)T], (24)；
T (i)=diag (S (kalarm,i) S(kalarm,i+1) … S(krelease,i-1)), (26)；
G(i,j)11=G (i, j) { 1:ny(krelease,i-kalarm,i),1}, (27)；
G(i,j)12=G (i, j) { 1:ny(krelease,i-kalarm,i),2:ny}, (28)；
Step 4.2：Set intermittent fault separation false segmentation rate
Step 4.3：Design intermittent fault separation threshold value JI(j)
Wherein,
Step 4.4：Intermittent fault separation is carried out according to following criterions
If TI(i, j) ＞ JI(j), then sensor j is faulty；
If TI(i,j)≤JI(j), then sensor j fault-frees；
Step 4.5：Calculate intermittent fault estimate
Wherein,
Advantageous effects caused by the present invention：
The redundancy of the invention fully excavated available for fault diagnosis, can be between real-time online detection, separation and estimation Have a rest sensor fault, independent of failure code table, effective guarantee bullet train critical system interval sensor fault diagnosis Practical application request.
Brief description of the drawings
Fig. 1 is the flow chart of the inventive method.
Fig. 2 is the damage curve schematic diagram of bullet train critical system interval sensor of the present invention.
Fig. 3 is the failure detection result schematic diagram of bullet train critical system interval sensor of the present invention.
Fig. 4 is the fault reconstruction result schematic diagram of bullet train critical system interval sensor of the present invention.
Fig. 5 is the Fault Estimation result schematic diagram of bullet train critical system interval sensor of the present invention.
Embodiment
Below in conjunction with the accompanying drawings and embodiment is described in further detail to the present invention：
A kind of robust interval sensor fault diagnosis method of bullet train critical system, its flow is as shown in figure 1, specific Comprise the following steps：
Step 1：Establish bullet train critical system discrete-time state-space model and identification model parameter
Wherein,Respectively system mode, control input, measurement output； Respectively process noise, measurement noise；For sensor fault； For systematic parameter； For parameter uncertainty；Meet following condition：Original state x (0) average, covariance, second moment are respectivelyP0, Σ0；Make an uproar Sound w (k), v (k) average are zero, and covariance matrix is respectively Σw(k), Σv(k)；Parameter uncertainty Aδ(k),Bδ(k),Cδ(k) Average be zero, covariance matrix is respectively
Step 2：Bullet train critical system robust Residual Generation device is designed, is specifically comprised the following steps：
Step 2.1：Off-line calculation gain matrix K (k)
K (k)=G (k) Cc(k)TQ(k)-1, (2)；
Wherein,
Rapid 2.2：In line computation robust residual error r (k)
Wherein,
Step 3：Bullet train critical system robust interval Transducer-fault Detecting Method is designed, specifically includes following step Suddenly：
Step 3.1：Calculate intermittent fault detection statistic TD(k)
Step 3.2：Set intermittent fault detection rate of false alarm Pfa
Pfa=αfa, (12)；
Step 3.3：Design intermittent fault detection threshold value JD
Wherein,
Step 3.4：Intermittent fault detection is carried out according to following criterions
If TD(k-1)≤JD,TD(k) ＞ JD, then the k moment break down, fault warning indicatrix Ia=1；
If TD(k-1) ＞ JD,TD(k)≤JD, then k moment failure vanishes, trouble shooting indicatrix Ir=1；
Correspondingly, the fault warning time of i-th intermittent fault and trouble shooting time are respectively
kalarm,i=min (k | TD(k) ＞ JD,k≥krelease,i-1+1), (15)；
krelease,i=min (k | TD(k)≤JD,k≥kalarm,i+1), (16)；
Failure is as shown in Fig. 2 failure detection result is as shown in Figure 3.
Step 4：Design the separation of bullet train critical system robust interval sensor fault and method of estimation, specifically include as Lower step：
Step 4.1：Calculate intermittent fault separation statistic TI(i,j)
Wherein,
L(i,j)11=G (i, j)11 TT(i)-1G(i,j)11, (18)；
L(i,j)12=G (i, j)11 TT(i)-1G(i,j)12, (19)；
L(i,j)21=G (i, j)12 TT(i)-1G(i,j)11, (20)；
L(i,j)22=G (i, j)12 TT(i)-1G(i,j)12, (21)；
d(i,j)1=G (i, j)11 TT(i)-1r(kalarm,i,krelease,i-1), (22)；
d(i,j)2=G (i, j)12 TT(i)-1r(kalarm,i,krelease,i-1), (23)；
r(kalarm,i,krelease,i- 1)=[r (kalarm,i)T r(kalarm,i+1)T … r(krelease,i-1)T], (24)；
T (i)=diag (S (kalarm,i) S(kalarm,i+1) … S(krelease,i-1)), (26)；
G(i,j)11=G (i, j) { 1:ny(krelease,i-kalarm,i),1}, (27)；
G(i,j)12=G (i, j) { 1:ny(krelease,i-kalarm,i),2:ny}, (28)；
Step 4.2：Set intermittent fault separation false segmentation rate
Step 4.3：Design intermittent fault separation threshold value JI(j)
Wherein,
Step 4.4：Intermittent fault separation is carried out according to following criterions
If TI(i, j) ＞ JI(j), then sensor j is faulty；
If TI(i,j)≤JI(j), then sensor j fault-frees；
Step 4.5：Calculate intermittent fault estimate
Wherein,
Fault reconstruction and Fault Estimation result difference are as shown in Figure 4, Figure 5.
Certainly, described above is not limitation of the present invention, and the present invention is also not limited to the example above, this technology neck The variations, modifications, additions or substitutions that the technical staff in domain is made in the essential scope of the present invention, it should also belong to the present invention's Protection domain.

Claims (1)

  1. A kind of 1. robust interval sensor fault diagnosis method of bullet train critical system, it is characterised in that：Specifically include as Lower step：
    Step 1：Establish bullet train critical system discrete-time state-space model and identification model parameter
    <mrow> <mtable> <mtr> <mtd> <mrow> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mi>c</mi> </msub> <mo>(</mo> <mi>k</mi> <mo>)</mo> <mo>+</mo> <msub> <mi>A</mi> <mi>&amp;delta;</mi> </msub> <mo>(</mo> <mi>k</mi> <mo>)</mo> <mo>)</mo> </mrow> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mi>B</mi> <mi>c</mi> </msub> <mo>(</mo> <mi>k</mi> <mo>)</mo> <mo>+</mo> <msub> <mi>B</mi> <mi>&amp;delta;</mi> </msub> <mo>(</mo> <mi>k</mi> <mo>)</mo> <mo>)</mo> </mrow> <mi>u</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>w</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>y</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>c</mi> </msub> <mo>(</mo> <mi>k</mi> <mo>)</mo> <mo>+</mo> <msub> <mi>C</mi> <mi>&amp;delta;</mi> </msub> <mo>(</mo> <mi>k</mi> <mo>)</mo> <mo>)</mo> </mrow> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>v</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
    Wherein,Respectively system mode, control input, measurement output； Respectively process noise, measurement noise；For sensor fault； For systematic parameter； For parameter uncertainty；Meet following condition：Original state x (0) average, covariance, second moment are respectivelyP0, Σ0；Make an uproar Sound w (k), v (k) average are zero, and covariance matrix is respectively Σw(k), Σv(k)；Parameter uncertainty Aδ(k),Bδ(k),Cδ(k) Average be zero, covariance matrix is respectively
    Step 2：Bullet train critical system robust Residual Generation device is designed, is specifically comprised the following steps：
    Step 2.1：Off-line calculation gain matrix K (k)
    K (k)=G (k) Cc(k)TQ(k)-1, (2)；
    Wherein,
    <mrow> <mi>G</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>A</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>P</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mi>A</mi> <mi>c</mi> </msub> <msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>+</mo> <msub> <mi>&amp;Sigma;</mi> <mrow> <msub> <mi>A</mi> <mi>&amp;delta;</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>&amp;Sigma;</mi> <mrow> <msub> <mi>B</mi> <mi>&amp;delta;</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>u</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>&amp;Sigma;</mi> <mrow> <mi>w</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>,</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
    <mrow> <mi>Q</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>C</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>G</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <msub> <mi>C</mi> <mi>c</mi> </msub> <msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>+</mo> <msub> <mi>&amp;Sigma;</mi> <mrow> <msub> <mi>C</mi> <mi>&amp;delta;</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <msub> <mi>&amp;Sigma;</mi> <mrow> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> </msub> </mrow> </msub> <mo>+</mo> <msub> <mi>&amp;Sigma;</mi> <mrow> <mi>v</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> </msub> <mo>,</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
    <mrow> <mi>P</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mrow> <msub> <mi>n</mi> <mi>x</mi> </msub> <mo>&amp;times;</mo> <msub> <mi>n</mi> <mi>x</mi> </msub> </mrow> </msub> <mo>-</mo> <mi>K</mi> <mo>(</mo> <mi>k</mi> <mo>)</mo> <msub> <mi>C</mi> <mi>c</mi> </msub> <mo>(</mo> <mi>k</mi> <mo>)</mo> <mo>)</mo> </mrow> <mi>G</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>,</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
    Rapid 2.2：In line computation robust residual error r (k)
    <mrow> <mi>r</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>y</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>C</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mover> <mi>x</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>,</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
    Wherein,
    <mrow> <mover> <mi>x</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mover> <mi>x</mi> <mo>&amp;OverBar;</mo> </mover> <mn>0</mn> </msub> <mo>,</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
    <mrow> <mover> <mi>x</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>A</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mover> <mi>x</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>B</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>u</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>,</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
    <mrow> <mi>r</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <mi>y</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>C</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mover> <mi>x</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>,</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
    <mrow> <mover> <mi>x</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <mover> <mi>x</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>+</mo> <mi>K</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>r</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>,</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
    Step 3：Bullet train critical system robust interval Transducer-fault Detecting Method is designed, is specifically comprised the following steps：
    Step 3.1：Calculate intermittent fault detection statistic TD(k)
    <mrow> <msub> <mi>T</mi> <mi>D</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>r</mi> <msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>T</mi> </msup> <msup> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mi>c</mi> </msub> <mo>(</mo> <mi>k</mi> <mo>)</mo> <mi>P</mi> <mo>(</mo> <mi>k</mi> <mo>)</mo> <msub> <mi>C</mi> <mi>c</mi> </msub> <msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>+</mo> <msub> <mi>&amp;Sigma;</mi> <mrow> <msub> <mi>C</mi> <mi>&amp;delta;</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>&amp;Sigma;</mi> <mrow> <mi>v</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> </msub> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mi>r</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>,</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
    Step 3.2：Set intermittent fault detection rate of false alarm Pfa
    Pfa=αfa, (12)；
    Step 3.3：Design intermittent fault detection threshold value JD
    <mrow> <msub> <mi>J</mi> <mi>D</mi> </msub> <mo>=</mo> <msubsup> <mi>&amp;chi;</mi> <msub> <mi>&amp;alpha;</mi> <mrow> <mi>f</mi> <mi>a</mi> </mrow> </msub> <mn>2</mn> </msubsup> <mo>,</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
    Wherein,
    <mrow> <mi>P</mi> <mo>{</mo> <msub> <mi>T</mi> <mi>D</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>&gt;</mo> <msubsup> <mi>&amp;chi;</mi> <msub> <mi>&amp;alpha;</mi> <mrow> <mi>f</mi> <mi>a</mi> </mrow> </msub> <mn>2</mn> </msubsup> <mo>|</mo> <mi>r</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>~</mo> <mi>N</mi> <mrow> <mo>(</mo> <msub> <mn>0</mn> <mrow> <msub> <mi>n</mi> <mi>y</mi> </msub> <mo>&amp;times;</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mi>c</mi> </msub> <mo>(</mo> <mi>k</mi> <mo>)</mo> <mi>P</mi> <mo>(</mo> <mi>k</mi> <mo>)</mo> <msub> <mi>C</mi> <mi>c</mi> </msub> <msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>+</mo> <msub> <mi>&amp;Sigma;</mi> <mrow> <msub> <mi>C</mi> <mi>&amp;delta;</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>&amp;Sigma;</mi> <mrow> <mi>v</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> </msub> <mo>)</mo> </mrow> <mo>}</mo> <mo>=</mo> <msub> <mi>&amp;alpha;</mi> <mrow> <mi>f</mi> <mi>a</mi> </mrow> </msub> <mo>,</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>14</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
    Step 3.4：Intermittent fault detection is carried out according to following criterions
    If TD(k-1)≤JD,TD(k) ＞ JD, then the k moment break down, fault warning indicatrix Ia=1；
    If TD(k-1) ＞ JD,TD(k)≤JD, then k moment failure vanishes, trouble shooting indicatrix Ir=1；
    Correspondingly, the fault warning time of i-th intermittent fault and trouble shooting time are respectively
    kalarm,i=min (k | TD(k) ＞ JD,k≥krelease,i-1+1), (15)；
    krelease,i=min (k | TD(k)≤JD,k≥kalarm,i+1), (16)；
    Step 4：The separation of bullet train critical system robust interval sensor fault and method of estimation are designed, specifically includes following step Suddenly：
    Step 4.1：Calculate intermittent fault separation statistic TI(i,j)
    <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>T</mi> <mi>I</mi> </msub> <mrow> <mo>(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mi>d</mi> <msub> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> <mn>1</mn> </msub> <mo>-</mo> <mi>L</mi> <msub> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> <mn>12</mn> </msub> <mi>L</mi> <msup> <msub> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> <mn>22</mn> </msub> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mi>d</mi> <msub> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&amp;times;</mo> <msup> <mrow> <mo>(</mo> <mi>L</mi> <msub> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> <mn>11</mn> </msub> <mo>-</mo> <mi>L</mi> <msub> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> <mn>12</mn> </msub> <mi>L</mi> <msup> <msub> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> <mn>22</mn> </msub> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mi>L</mi> <msub> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> <mn>21</mn> </msub> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&amp;times;</mo> <mrow> <mo>(</mo> <mi>d</mi> <msub> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> <mn>1</mn> </msub> <mo>-</mo> <mi>L</mi> <msub> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> <mn>12</mn> </msub> <mi>L</mi> <msup> <msub> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> <mn>22</mn> </msub> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mi>d</mi> <msub> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>17</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
    Wherein,
    L(i,j)11=G (i, j)11 TT(i)-1G(i,j)11, (18)；
    L(i,j)12=G (i, j)11 TT(i)-1G(i,j)12, (19)；
    L(i,j)21=G (i, j)12 TT(i)-1G(i,j)11, (20)；
    L(i,j)22=G (i, j)12 TT(i)-1G(i,j)12, (21)；
    d(i,j)1=G (i, j)11 TT(i)-1r(kalarm,i,krelease,i-1), (22)；
    d(i,j)2=G (i, j)12 TT(i)-1r(kalarm,i,krelease,i-1), (23)；
    r(kalarm,i,krelease,i- 1)=[r (kalarm,i)T r(kalarm,i+1)T…r(krelease,i-1)T], (24)；
    <mrow> <mi>G</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>e</mi> <mrow> <mi>c</mi> <mi>o</mi> <mi>l</mi> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mrow> <msub> <mi>n</mi> <mi>y</mi> </msub> <mo>&amp;times;</mo> <msub> <mi>n</mi> <mi>y</mi> </msub> </mrow> </msub> <mo>+</mo> <msub> <mi>C</mi> <mi>c</mi> </msub> <mo>(</mo> <msub> <mi>k</mi> <mrow> <mi>a</mi> <mi>l</mi> <mi>a</mi> <mi>r</mi> <mi>m</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>)</mo> <mi>Z</mi> <mo>(</mo> <msub> <mi>k</mi> <mrow> <mi>a</mi> <mi>l</mi> <mi>a</mi> <mi>r</mi> <mi>m</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>)</mo> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>e</mi> <mrow> <mi>c</mi> <mi>o</mi> <mi>l</mi> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mrow> <msub> <mi>n</mi> <mi>y</mi> </msub> <mo>&amp;times;</mo> <msub> <mi>n</mi> <mi>y</mi> </msub> </mrow> </msub> <mo>+</mo> <msub> <mi>C</mi> <mi>c</mi> </msub> <mo>(</mo> <mrow> <msub> <mi>k</mi> <mrow> <mi>a</mi> <mi>l</mi> <mi>a</mi> <mi>r</mi> <mi>m</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> <mi>Z</mi> <mo>(</mo> <mrow> <msub> <mi>k</mi> <mrow> <mi>a</mi> <mi>l</mi> <mi>a</mi> <mi>r</mi> <mi>m</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>e</mi> <mrow> <mi>c</mi> <mi>o</mi> <mi>l</mi> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mrow> <msub> <mi>n</mi> <mi>y</mi> </msub> <mo>&amp;times;</mo> <msub> <mi>n</mi> <mi>y</mi> </msub> </mrow> </msub> <mo>+</mo> <msub> <mi>C</mi> <mi>c</mi> </msub> <mo>(</mo> <mrow> <msub> <mi>k</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>l</mi> <mi>e</mi> <mi>a</mi> <mi>s</mi> <mi>e</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> <mi>Z</mi> <mo>(</mo> <mrow> <msub> <mi>k</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>l</mi> <mi>e</mi> <mi>a</mi> <mi>s</mi> <mi>e</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>25</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
    T (i)=diag (S (kalarm,i) S(kalarm,i+1) … S(krelease,i-1)), (26)；
    G(i,j)11=G (i, j) { 1:ny(krelease,i-kalarm,i),1}, (27)；
    G(i,j)12=G (i, j) { 1:ny(krelease,i-kalarm,i),2:ny}, (28)；
    Step 4.2：Set intermittent fault separation false segmentation rate
    <mrow> <msub> <mi>P</mi> <msubsup> <mi>f</mi> <mi>i</mi> <mrow> <mo>(</mo> <mi>j</mi> <mo>)</mo> </mrow> </msubsup> </msub> <mo>=</mo> <msub> <mi>&amp;alpha;</mi> <msubsup> <mi>f</mi> <mi>i</mi> <mrow> <mo>(</mo> <mi>j</mi> <mo>)</mo> </mrow> </msubsup> </msub> <mo>,</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>29</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
    Step 4.3：Design intermittent fault separation threshold value JI(j)
    <mrow> <mi>P</mi> <mo>{</mo> <msub> <mi>T</mi> <mi>I</mi> </msub> <mrow> <mo>(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>)</mo> </mrow> <mo>&gt;</mo> <msubsup> <mi>&amp;chi;</mi> <msub> <mi>&amp;alpha;</mi> <mrow> <mi>f</mi> <mi>a</mi> </mrow> </msub> <mn>2</mn> </msubsup> <mo>|</mo> <msubsup> <mi>f</mi> <mi>i</mi> <mrow> <mo>(</mo> <mi>j</mi> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <mn>0</mn> <mo>}</mo> <mo>=</mo> <msub> <mi>&amp;alpha;</mi> <msubsup> <mi>f</mi> <mi>i</mi> <mrow> <mo>(</mo> <mi>j</mi> <mo>)</mo> </mrow> </msubsup> </msub> <mo>,</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>30</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
    Wherein,
    <mrow> <msub> <mi>J</mi> <mi>I</mi> </msub> <mrow> <mo>(</mo> <mi>j</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>&amp;alpha;</mi> <msubsup> <mi>f</mi> <mi>i</mi> <mrow> <mo>(</mo> <mi>j</mi> <mo>)</mo> </mrow> </msubsup> </msub> <mo>,</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>31</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
    Step 4.4：Intermittent fault separation is carried out according to following criterions
    If TI(i, j) ＞ JI(j), then sensor j is faulty；
    If TI(i,j)≤JI(j), then sensor j fault-frees；
    Step 4.5：Calculate intermittent fault estimate
    <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mi>f</mi> <mo>^</mo> </mover> <mi>i</mi> </msub> <mo>=</mo> <mi>d</mi> <mi>i</mi> <mi>a</mi> <mi>g</mi> <mo>{</mo> <msup> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msub> <mn>1</mn> <msup> <mi>R</mi> <mo>+</mo> </msup> </msub> <mrow> <mo>(</mo> <mi>&amp;xi;</mi> <mo>(</mo> <mrow> <mi>i</mi> <mo>,</mo> <mn>1</mn> </mrow> <mo>)</mo> <mo>-</mo> <msub> <mi>J</mi> <mi>I</mi> </msub> <mo>(</mo> <mn>1</mn> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msub> <mn>1</mn> <msup> <mi>R</mi> <mo>+</mo> </msup> </msub> <mrow> <mo>(</mo> <mi>&amp;xi;</mi> <mo>(</mo> <mrow> <mi>i</mi> <mo>,</mo> <mn>2</mn> </mrow> <mo>)</mo> <mo>-</mo> <msub> <mi>J</mi> <mi>I</mi> </msub> <mo>(</mo> <mn>2</mn> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mrow> <msub> <mn>1</mn> <msup> <mi>R</mi> <mo>+</mo> </msup> </msub> <mrow> <mo>(</mo> <mi>&amp;xi;</mi> <mo>(</mo> <mrow> <mi>i</mi> <mo>,</mo> <msub> <mi>n</mi> <mi>y</mi> </msub> </mrow> <mo>)</mo> <mo>-</mo> <msub> <mi>J</mi> <mi>I</mi> </msub> <mo>(</mo> <msub> <mi>n</mi> <mi>y</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mi>T</mi> </msup> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&amp;times;</mo> <mi>arg</mi> <munder> <mi>max</mi> <mi>&amp;beta;</mi> </munder> <mi>ln</mi> <mrow> <mo>(</mo> <mi>p</mi> <mo>(</mo> <mrow> <mi>r</mi> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mrow> <mi>a</mi> <mi>l</mi> <mi>a</mi> <mi>r</mi> <mi>m</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>,</mo> <mi>r</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>k</mi> <mrow> <mi>a</mi> <mi>l</mi> <mi>a</mi> <mi>r</mi> <mi>m</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>r</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>k</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>l</mi> <mi>e</mi> <mi>a</mi> <mi>s</mi> <mi>e</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>|</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>=</mo> <mi>&amp;beta;</mi> </mrow> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mi>d</mi> <mi>i</mi> <mi>a</mi> <mi>g</mi> <mo>{</mo> <msup> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msub> <mn>1</mn> <msup> <mi>R</mi> <mo>+</mo> </msup> </msub> <mrow> <mo>(</mo> <mi>&amp;xi;</mi> <mo>(</mo> <mrow> <mi>i</mi> <mo>,</mo> <mn>1</mn> </mrow> <mo>)</mo> <mo>-</mo> <msub> <mi>J</mi> <mi>I</mi> </msub> <mo>(</mo> <mn>1</mn> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msub> <mn>1</mn> <msup> <mi>R</mi> <mo>+</mo> </msup> </msub> <mrow> <mo>(</mo> <mi>&amp;xi;</mi> <mo>(</mo> <mrow> <mi>i</mi> <mo>,</mo> <mn>2</mn> </mrow> <mo>)</mo> <mo>-</mo> <msub> <mi>J</mi> <mi>I</mi> </msub> <mo>(</mo> <mn>2</mn> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mrow> <msub> <mn>1</mn> <msup> <mi>R</mi> <mo>+</mo> </msup> </msub> <mrow> <mo>(</mo> <mi>&amp;xi;</mi> <mo>(</mo> <mrow> <mi>i</mi> <mo>,</mo> <msub> <mi>n</mi> <mi>y</mi> </msub> </mrow> <mo>)</mo> <mo>-</mo> <msub> <mi>J</mi> <mi>I</mi> </msub> <mo>(</mo> <msub> <mi>n</mi> <mi>y</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mi>T</mi> </msup> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&amp;times;</mo> <mi>S</mi> <msup> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mrow> <mi>a</mi> <mi>l</mi> <mi>a</mi> <mi>r</mi> <mi>m</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>k</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>l</mi> <mi>e</mi> <mi>a</mi> <mi>s</mi> <mi>e</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mi>h</mi> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mrow> <mi>a</mi> <mi>l</mi> <mi>a</mi> <mi>r</mi> <mi>m</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>k</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>l</mi> <mi>e</mi> <mi>a</mi> <mi>s</mi> <mi>e</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>32</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
    Wherein,
    <mrow> <mi>S</mi> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mrow> <mi>a</mi> <mi>l</mi> <mi>a</mi> <mi>r</mi> <mi>m</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>k</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>l</mi> <mi>e</mi> <mi>a</mi> <mi>s</mi> <mi>e</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>l</mi> <mo>=</mo> <msub> <mi>k</mi> <mrow> <mi>a</mi> <mi>l</mi> <mi>a</mi> <mi>r</mi> <mi>m</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>k</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>l</mi> <mi>e</mi> <mi>a</mi> <mi>s</mi> <mi>e</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>-</mo> <mn>1</mn> </mrow> </munderover> <msup> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mrow> <msub> <mi>n</mi> <mi>y</mi> </msub> <mo>&amp;times;</mo> <msub> <mi>n</mi> <mi>y</mi> </msub> </mrow> </msub> <mo>+</mo> <msub> <mi>C</mi> <mi>c</mi> </msub> <mo>(</mo> <mi>l</mi> <mo>)</mo> <mi>Z</mi> <mo>(</mo> <mi>l</mi> <mo>)</mo> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mi>Q</mi> <msup> <mrow> <mo>(</mo> <mi>l</mi> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mrow> <msub> <mi>n</mi> <mi>y</mi> </msub> <mo>&amp;times;</mo> <msub> <mi>n</mi> <mi>y</mi> </msub> </mrow> </msub> <mo>+</mo> <msub> <mi>C</mi> <mi>c</mi> </msub> <mo>(</mo> <mi>l</mi> <mo>)</mo> <mi>Z</mi> <mo>(</mo> <mi>l</mi> <mo>)</mo> <mo>)</mo> </mrow> <mo>,</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>33</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
    <mrow> <mi>h</mi> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mrow> <mi>a</mi> <mi>l</mi> <mi>a</mi> <mi>r</mi> <mi>m</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>k</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>l</mi> <mi>e</mi> <mi>a</mi> <mi>s</mi> <mi>e</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>l</mi> <mo>=</mo> <msub> <mi>k</mi> <mrow> <mi>a</mi> <mi>l</mi> <mi>a</mi> <mi>r</mi> <mi>m</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>k</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>l</mi> <mi>e</mi> <mi>a</mi> <mi>s</mi> <mi>e</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>-</mo> <mn>1</mn> </mrow> </munderover> <msup> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mrow> <msub> <mi>n</mi> <mi>y</mi> </msub> <mo>&amp;times;</mo> <msub> <mi>n</mi> <mi>y</mi> </msub> </mrow> </msub> <mo>+</mo> <msub> <mi>C</mi> <mi>c</mi> </msub> <mo>(</mo> <mi>l</mi> <mo>)</mo> <mi>Z</mi> <mo>(</mo> <mi>l</mi> <mo>)</mo> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mi>Q</mi> <msup> <mrow> <mo>(</mo> <mi>l</mi> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mi>r</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>)</mo> </mrow> <mo>,</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>34</mn> <mo>)</mo> </mrow> <mo>.</mo> </mrow>
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